The fundamental difference between shear alpha viscosity and turbulent magnetorotational stresses

Numerical simulations of turbulent, magnetized, differentially rotating flows driven by the magnetorotational instability (MRI) are often used to calculate the effective values of alpha viscosity that is invoked in analytical models of accretion discs. In this paper, we use various dynamical models of turbulent magnetohydrodynamic stresses, as well as numerical simulations of shearing boxes, to show that angular momentum transport in MRI-driven accretion discs cannot be described by the standard model for shear viscosity. In particular, we demonstrate that turbulent magnetorotational stresses are not linearly proportional to the local shear and vanish identically for angular velocity profiles that increase outwards.